Vector. What is a vector?
Such a concept as a vector is considered in almost all natural sciences, and it can have completely different meanings, therefore it is impossible to give a unique definition of a vector for all fields. But let's try to figure it out. So what is a vector?
The concept of vector in classical geometry
A vector in geometry is a segment for which it is indicated which of its points is the beginning and which is the end. That is, to put it more simply, a vector is a directed segment.
Accordingly, the vector is denoted (what is considered above), as well as the segment, that is, in two capital letters of the Latin alphabet with the addition of a line or arrow pointing to the right from above. You can also sign it with a lowercase (small) letter of the Latin alphabet with a dash or arrow. The arrow is always directed to the right and does not change depending on the location of the vector.
Thus, the vector has direction and length.
The designation of the vector contains its direction. This is expressed as in the figure below.
Changing direction changes the value of the vector to the opposite.
The length of a vector is the length of the segment from which it is formed. It is denoted as a module of a vector. This is shown in the figure below.
Accordingly, zero is a vector whose length is zero. From this it follows that the zero vector is a point, and the points of beginning and end coincide in it.
The length of the vector - the value is always not negative. In other words, if there is a segment, then it necessarily has a certain length or is a point, then its length is zero.
The very notion of a point is basic and has no definition.
There are special formulas and rules for vectors with which you can perform addition.
Triangle rule. To add vectors according to this rule, it is enough to combine the end of the first vector and the beginning of the second, using parallel translation, and join them. The resulting third vector will be equal to the addition of the other two.
Parallelogram rule. For addition according to this rule, it is necessary to draw both vectors from one point, and then draw another vector from the end of each one. That is, the second will be drawn from the first vector, and the first from the second.The result is a new intersection point and a parallelogram is formed. If we combine the intersection point of the beginnings and ends of the vectors, then the resulting vector will be the result of addition.
Similarly, it is possible to perform and subtract.
Similar to adding vectors, it is possible to subtract them. It is based on the principle shown in the figure below.
That is, it is sufficient to represent the subtracted vector in the form of a vector, the opposite to it, and to calculate by the principles of addition.
Also absolutely any non-zero vector can be multiplied by any number k, this will change its length k times.
In addition to these, there are other formulas of vectors (for example, to express the length of a vector through its coordinates).
Location of vectors
Surely, many were faced with such a concept as a collinear vector. What is collinearity?
The collinearity of vectors is the equivalent of parallel lines. If two vectors lie on straight lines that are parallel to each other, or on one straight line, then such vectors are called collinear.
Direction. With respect to each other, collinear vectors can be co-directed or opposite-directed, this is determined by the direction of the vectors.Accordingly, if the vector is co-directed with another, then the vector opposite to it is oppositely directed.
The first figure shows two oppositely directed vectors and the third one, which is not collinear to them.
After the introduction of the above properties, it is possible to give a definition to equal vectors - these are vectors that are directed in the same direction and have the same length of segments from which they are formed.
In many sciences, the concept of a radius vector is also used. Such a vector describes the position of one point of the plane relative to another fixed point (often this is the origin).
Vectors in physics
Suppose when solving a problem a condition arose: the body moves at a speed of 3 m / s. This means that the body moves with a specific direction in one straight line, so this variable will be a vector value. For the solution, it is important to know both the value and the direction, since depending on the consideration the speed can be equal to 3 m / s and -3 m / s.
In general, the vector in physics is used to indicate the direction of the force acting on the body, and to determine the resultant.
When specifying these forces in the figure, they are indicated by arrows with the signature of the vector above it.Classically, the length of the arrow is just as important, with the help of which they indicate which force acts stronger, but this side property is not worth relying on it.
Vector in linear algebra and mathematical analysis
Elements of linear spaces are also called vectors, but in this case they are an ordered system of numbers describing some of the elements. Therefore, the direction in this case no longer has any importance. The definition of a vector in classical geometry and in mathematical analysis is very different.
Projected vector - what is it?
Quite often, for correct and convenient calculation, it is necessary to expand the vector in two-dimensional or three-dimensional space along the axes of coordinates. This operation is necessary, for example, in mechanics when calculating the forces acting on the body. Vector in physics is used quite often.
To perform a projection, it is sufficient to lower the perpendiculars from the beginning and end of the vector onto each of the coordinate axes, the segments obtained on them will be called the projection of the vector onto the axis.
To calculate the length of the projection, it is sufficient to multiply its initial length by a certain trigonometric function, which is obtained by solving a mini-problem.